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A130853 Runs of 1's of lengths 1, Fibonacci numbers F(1), F(2), F(3), ... (A000045) separated by 0's.

%I A130853 #45 May 22 2025 01:03:19
%S A130853 0,1,0,1,0,1,1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,
%T A130853 1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,
%U A130853 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1
%N A130853 Runs of 1's of lengths 1, Fibonacci numbers F(1), F(2), F(3), ... (A000045) separated by 0's.
%C A130853 Might be called a Fibonacci message.
%H A130853 Antti Karttunen, <a href="/A130853/b130853.txt">Table of n, a(n) for n = 1..46390</a>
%H A130853 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A130853 a(n) = b(n+1) - b(n) where b(n) = round(LambertW((phi^(3/2 + n)*log(phi))/sqrt(5)) / log(phi)), phi = (1 + sqrt(5))/2. - _Alan Michael Gómez Calderón_, Dec 11 2024
%e A130853 Begin with 0. First Fibonacci number F(1)=1, so append 1's to 0 once - 01, append 0 - 010, F(2)=1, append 1's once and 0 - 01010, F(3)=2, we append two 1's and 0 - 01010110, ...
%e A130853 As a triangle:
%e A130853   0, 1;
%e A130853   0, 1;
%e A130853   0, 1, 1;
%e A130853   0, 1, 1, 1;
%e A130853   0, 1, 1, 1, 1, 1;
%e A130853   0, 1, 1, 1, 1, 1, 1, 1, 1;
%e A130853   0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A130853   0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A130853   ...
%p A130853 ts_Finonacci_zap:=proc(n) local i,j,tren,ans; ans := [ 0 ]: for i from 1 to n do tren := combinat[fibonacci](i): for j from 1 to tren do ans:=[ op(ans), 1 ]: od: ans:=[ op(ans), 0 ]: od; RETURN(ans) end: ts_Finonacci_zap(16);
%p A130853 # second Maple program:
%p A130853 T:= n-> [0,1$(<<0|1>, <1|1>>^n)[1,2]][]:
%p A130853 seq(T(n), n=1..10);  # _Alois P. Heinz_, Dec 11 2024
%t A130853 T[n_] := Join[{0}, Table[1, MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]]]];
%t A130853 Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, May 21 2025, after _Alois P. Heinz_ *)
%o A130853 (PARI) { n=0; i=0; while(n<22, n++; i++; write("b130853.txt", i, " ", 0); k = fibonacci(n); while(k>0, i++; k--; write("b130853.txt", i, " ", 1))); }; \\ _Antti Karttunen_, Dec 07 2017
%Y A130853 Cf. A000045, A093521, A232896 (the positions of zeros).
%Y A130853 Cf. A001622, A003849, A005614, A194029.
%K A130853 nonn,tabf
%O A130853 1,1
%A A130853 _Jani Melik_, Jul 21 2007